Understanding the resilience of infrastructures such as transportation network has significant importance for our daily life. Recently, a homogeneous spatial network model was developed for studying spatial embedded networks with characteristic link length such as power-grids and the brain. However, although many real-world networks are spatially embedded and their links have characteristics length such as pipelines, power lines or ground transportation lines they are not homogeneous but rather heterogeneous. For example, density of links within cities are significantly higher than between cities. Here we present and study numerically and analytically a similar realistic heterogeneous spatial modular model using percolation process to better understand the effect of heterogeneity on such networks. The model assumes that inside a city there are many lines connecting different locations, while long lines between the cities are sparse and usually directly connecting only a few nearest neighbours cities in a two dimensional plane. We find that this model experiences two distinct continues transitions, one when the cities disconnect from each other and the second when each city breaks apart. Although the critical threshold for site percolation in 2D grid remains an open question we analytically find the critical threshold for site percolation in this model. In addition, while the homogeneous model experience a single transition having a unique phenomenon called \textit{critical stretching} where a geometric crossover from random to spatial structure in different scales found to stretch non-linearly with the characteristic length at criticality. Here we show that the heterogeneous model does not experience such a phenomenon indicating that critical stretching strongly depends on the network structure.

中文翻译：

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空间模块化网络的两个转变

了解交通网络等基础设施的弹性对我们的日常生活具有重要意义。最近，开发了一种同构空间网络模型，用于研究具有特征链路长度的空间嵌入式网络，例如电网和大脑。然而，尽管许多现实世界的网络是空间嵌入的，并且它们的链接具有特征长度，例如管道、电力线或地面运输线，但它们不是同质的，而是异质的。例如，城市内部的联系密度明显高于城市之间的联系。在这里，我们使用渗流过程对类似的现实异构空间模块化模型进行数值和分析研究，以更好地了解异质性对此类网络的影响。该模型假设在一个城市内部有许多连接不同位置的线，而城市之间的长线是稀疏的，并且通常只直接连接二维平面上几个最近的相邻城市。我们发现该模型经历了两种截然不同的持续转变，一种是城市相互断开，另一种是每个城市分裂。尽管二维网格中站点渗透的临界阈值仍然是一个悬而未决的问题，但我们通过分析找到了该模型中站点渗透的临界阈值。此外，虽然同质模型经历了具有称为 \textit{临界拉伸} 的独特现象的单一过渡，其中发现从随机到不同尺度的空间结构的几何交叉发现在临界点与特征长度非线性地拉伸。